Physical Oceanography, Vol.
6, March, 2012 (Ukrainian Original No.
6, November–December, 2011)
IDENTIFICATION OF THE INITIAL FIELD OF CONCENTRATIONS OF C
IN THE BLACK SEA AFTER THE CHERNOBYL ACCIDENT ON THE BASIS OF
THE SOLUTION OF A DUAL PROBLEM
S. V. Kochergin
and V. S. Kochergin
By the method of dual equations, we identify the initial field of the concentrations of
in the Black
Sea after the Chernobyl accident. We determine the zones of possible radioactive precipitations and
compute the concentration of
in this region. It is shown that the results of numerical simulations
performed by using the obtained initial field is in good agreement with the data of measurements.
Keywords: identification, dual problem, transport model, Black Sea.
The solution of various ecological problems connected with the analysis of possible consequences of the
transportation of passive admixtures requires the investigation of the concentration fields formed under the in-
fluence of natural or technogenic sources. The process of formation of these fields and their dynamics can be
studied by the methods of mathematical simulation. In this case, it is necessary to solve two correlated prob-
lems. The first problem is to construct or choose an optimal transport model for an admixture taking into ac-
count the nature of its behavior in the marine media. The second problem is to specify the required input pa-
rameters of the transport model with regard for the available data of measurements. As the input parameters, we
take the coefficients of model equations, the velocity fields, and the initial data.
In the solution of specific ecological problems, it is often necessary to monitor not the space structures of
the fields of concentrations of the analyzed admixtures but the values of certain functionals in the investigated
area. Thus, we can consider the mean or total concentrations of certain admixtures in the investigated area.
These values of functionals can be found directly according to the values of concentration at the nodes of the
computational grid. In this case, the values attained at end of the period of integration are determined as a result
of the solution of the transport problem with known initial data and the functions of the sources of pollution.
Thus, by setting different input data, we obtain a series of concentration fields for which we compute the re-
quired functional in the analyzed area. Among these calculations, we can choose the optimal version according
to the condition that the functional should not exceed certain admissible values. On the other hand, the solution
of problems of this sort can be simplified by using the so-called dual equations. In this case, according to ,
the dual problem is solved only once and the values of a functional can be found for various initial fields. The
extrema of the solutions of the dual problem indicate the areas in the region of integration whose influence on
the values of the analyzed functional is maximum in the chosen time interval.
As a model in a domain
= D × [0, t ]
, we take the following equation:
Marine Hydrophysical Institute, Ukrainian National Academy of Sciences, Sevastopol, Ukraine.
Corresponding author; e-mail: email@example.com.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
36–42, November–December, 2011. Original article submitted January 21,
2010; revision submitted June 29, 2010.
0928–5105/12/2106–0401 © 2012 Springer Science+Business Media, Inc. 401